Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number λω is zero and the Weissenberg number λγ˙0 is above unity), (ii) nonlinear viscoelasticity (where both λω and λγ˙0 exceed unity), and (iii) linear viscoelasticity (where λω exceeds unity and where λγ˙0 approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.


I. INTRODUCTION
A rigid dumbbell has both length and width, the two key geometric properties of a polymer molecule, and these properties confer upon the polymer orientation (see Figures 1 and 2). By far, the most common experiment for probing the structural dynamics of polymeric liquids is oscillatory shear flow, 1-6 and when performed at large-amplitude (Refs. 7-12; see also Section 2 of Ref. 13), this test is particularly informative. In our previous work, we have used the diffusion equation for molecular orientation to derive the expression for the orientation distribution function by expanding it in powers of the shear rate amplitude. [37][38][39][40][41][42] This expression contains zeroth, first, second, third, and fourth harmonics. We learned that the zeroth order term in the expansion contributes only to the zeroth harmonic, the first order term to the first, the second order term to the zeroth and second, third order term to the first and third, and finally, the fourth order term to the zeroth, second, and fourth harmonics.  Table I classifies the literature on analytical solutions for orientation distribution in largeamplitude oscillatory shear flow. The connection between the components of the extra stress tensor and the orientation distribution function was established long ago, 28 and from this work, we know that only certain parts of the orientation distribution function contribute to each component of the extra stress tensor in simple shear flows, and that certain parts make no contribution. In our previous work, we have used the contributing parts to derive expressions for the shear stress and for both normal stress difference responses (see Eq. (82) of Ref. 37 and Eqs. (13) and (21) of Ref. 39). In our most recent work, we derived the full orientation distribution function, including the terms that make no contribution to the rheological responses [40][41][42] (see column titled "non-contributing terms" of Table I).
In this paper, we decompose the expression for the orientation distribution function into its Fourier components (putting the Fourier coefficients under common denominators). Then, by examining each Fourier component, we see the frequency content of each of the molecular motions and examine the orientation distribution directly, by deriving expressions for the zeroth, first, second, third, and fourth harmonics of the orientation in large-amplitude oscillatory shear flow, as functions of the dimensionless frequency, kx, and dimensionless shear rate amplitude, k_ c 0 . Our analytical solution, presented with three-dimensional color imagery, gives us insight into the molecular behaviors underpinning the observed higher harmonics in the nonlinear rheological responses. 14 We thus derive new structure-property relations for the orientation distribution in large-amplitude oscillatory shear flow. The rigid dumbbell suspension seems to mimic the behavior of polymer solutions and melts, at least qualitatively, and this seems to be because the overall orientation of the polymer chains gives the longest time constant for the polymeric liquids.
This paper concerns itself with the orientation distribution function of a suspension of rigid dumbbells, when the flow is given by a velocity field v x ðy; tÞ. We further restrict this paper to the periodic response, reached several cycles after startup of oscillatory shear, called alternance (see Section 6 of Refs. 15; 16). This work is an extension of recent articles, 37,38 wherein part of the orientation distribution is derived using the method of Bird and Armstrong 17 (see also Problem 11C.1 of Ref. 18).

A. The flow field
We are concerned here with the particular case of a homogeneous shearing flow (see Figure 1) called simple shear flow. The symbols used here and elsewhere are defined, along with their dimensions, in Tables II and III. The coefficient of proportionality between the velocity and position is _ c 0 f ðtÞ, where the special variable f depends solely on the time. This situation can be obtained by having the flow take place between two plates placed extremely close to one another. 19,20 We consider only the case where f ðtÞ ¼ cos xt so that Eq. (1) becomes 21,22 v which we call oscillatory shear flow. That is, the shear rate may be written as k_ cðtÞ ¼ k_ c 0 cos kxðt=kÞ; in which the dimensionless shear rate k_ c 0 (the Weissenberg number) and the dimensionless frequency kx (the Deborah number), appear. 16,21,22 All orientation distribution responses calculated herein are thus referred to this cosinusoidal shear rate (see Section V of Ref. 23). These and for rigid multibead-rods: where N is the number of equally spaced beads on the rod of length L, T is absolute temperature, k is Boltzmann's constant, f is the bead friction factor, given by Stokes' law, and b is the diameter of each of the beads of the dumbbell (see Figure 2). Equation (3) applies for continuum, and thus cannot apply when the gap, h, approaches the size of the molecules, L, or when L=h ( 1. For a detailed study of small-amplitude oscillatory shear for a rigid dumbbell suspension in confined gaps, see Section 5 of Ref. 24 or Section 2.3.3 of Ref. 25. In this paper, we consider the special case of Eq. (3) where higher harmonics arise in both the shear stress and in the normal stress difference responses.
Bead center to center length of rigid dumbbell L L Bead diameter L b Bead friction coefficient M=t f 3pbg s Boltzmann constant ML 2 =t 2 T=dumbbell k Cartesian coordinate, distance from stationary plate (Figure 1) L y Cartesian coordinate, flow direction ( Figure 1) L x Cartesian coordinate, transverse to flow direction ( Figure 1) where k_ c is the dimensionless oscillatory shear rate [given by Eq.
Also, the symbol for the distribution function used here is w that used in the second edition of Dynamics of Polymeric Liquids 27 is f . The dumbbell orientation is thus given in terms of the spherical coordinates h and / which are defined in Figure 2. When the fluid is at rest, all orientations are equally likely, and w is equal to a normalization constant, 1=4p, so that In other words, when the fluid is at rest, wðh; /Þ is spherically symmetric. Although the orientation distribution function for rigid dumbbell suspension has not been measured directly, its implications for rheological behaviors have been verified experimentally. Reasonable qualitative behaviors for oscillatory shear flow are obtained for amplitudes both small 26 and large. 37,38

II. METHOD
Schmalzer and Giacomin 40-42 recently solved Eq. (6) for large-amplitude oscillatory shear flow, including terms that do not contribute to the rheological responses, using the method of Bird and Armstrong. 17 This solution has the form where w 1 through w 4 are given by Eqs. (36), (44), (52), and (61) of Ref. 40. In this paper, we undertake the laborious Fourier decomposition of these expressions for w 1 through w 4 , putting the resulting Fourier coefficients under common denominators. By Fourier decomposition, we mean rewriting Eq. (9) as the Fourier series wðh; /; tÞ ¼ X 1 n¼0 ½w 0 n cos nxt þ w 00 n sin nxt; where w 0 n and w 00 n are the Fourier coefficients of the orientation distribution function respectively in-phase and out-of-phase with cos nxt. This Fourier decomposition, and finding of common denominators is straightforward, but tedious. To prevent introducing errors, we thus checked each result given below numerically by comparing the expressions before and after the Fourier decomposition to 16 significant figures. Specifically, Eq. (9) with Eqs.

III. RESULTS
From the results given below, we see that in oscillatory shear flow the polymer orientation distribution decomposes into both even (starting with the zeroth) and odd harmonics. We identify the even harmonics of the polymer motion with the differences in the normal components of the extra stress tensor, whose Fourier components are also all even. We can further identify the odd harmonics of the orientation with the yx-component of the extra stress tensor (called the shear stress), whose Fourier components are also all odd valued. To help the reader navigate

A. Zeroth harmonic
The Fourier coefficient of the zeroth harmonic of the orientation distribution function for a rigid dumbbell suspension in large-amplitude oscillatory shear flow is given by    11), as we progress from nonlinear steady shear flow to nonlinear viscoelasticity and to linear viscoelasticity (see this progression illustrated in Figure 3). In Figures 4  and 5, we see the origins of the peanut-shapes that we find in the overall orientation distribution (Figures 19 and 20). The sphericity of Figure 6 reflects the close proximity of the polymeric liquid structure in the linear viscoelastic region to the polymeric liquid at rest.

B. First harmonic
The Fourier coefficient of the first harmonic of orientation distribution function for a rigid dumbbell suspension in large-amplitude oscillatory shear flow is given by       12) and (13), as we progress from nonlinear steady shear flow to nonlinear viscoelasticity and to linear viscoelasticity (see this progression illustrated in Figure 3). In Figure 9 (Multimedia view), we see the only slight disturbance of the spherical orientation distribution that exists for a polymeric liquid at rest. Furthermore, comparing

C. Second harmonic
The Fourier coefficient of the second harmonic of the orientation distribution function for a rigid dumbbell suspension in large-amplitude oscillatory shear flow is given by     Figures 10 (Multimedia view)-12 (Multimedia view) illustrate the orientation distribution associated with the polymer motions arising at twice the test frequency, 2x, which are induced by the oscillatory shear deformation, and predicted by Eqs. (14) and (15), as we progress from nonlinear steady shear flow to nonlinear viscoelasticity and to linear viscoelasticity (see this progression illustrated in Figure 3). Comparing Figure 10 (Multimedia view) with Figure 19, or Figure 11 (Multimedia view) with Figure 20, we learn that the tilt of the peanuts in the overall orientation distributions for nonlinear behaviors is mainly due to the second harmonic. Furthermore, we see that the second harmonic contributes to orientation along the axes, whereas the first harmonic predicts polymer orientation away from the axes in the x; y-plane.   16) and (17), as we progress from nonlinear steady shear flow to nonlinear viscoelasticity and to linear viscoelasticity (see this progression illustrated in Figure 3). In Figure 15 (Multimedia view), we see the only slight disturbance of the spherical orientation distribution that exists for a polymeric liquid at rest. Furthermore, comparing Figure 13 (Multimedia view) with Figure 19, or Figure 15 (Multimedia view) with Figure 20, we learn that even for the nonlinear behaviors, the third harmonic is hardly reshaping the orientation distribution established by the zeroth harmonic. So the tilting of the peanut-shapes in Figures 19 and 20 is not caused by the first harmonic. Finally, comparing results for the first harmonic (Figures 7 (Multimedia view)-9 (Multimedia view)) with results for the third (Figure 13 (Multimedia view)-15 (Multimedia view)), we see that so that the slight contributions to orientation made with the first and third harmonics seem to be, at least loosely, linked by a h-rotation.

E. Fourth harmonic
The Fourier coefficient of the fourth harmonic of orientation distribution function for a rigid dumbbell suspension in large-amplitude oscillatory shear flow is given by and  (19) and (20), as we progress from nonlinear steady shear flow to nonlinear viscoelasticity and to linear viscoelasticity (see this progression illustrated in Figure 3). Figures 16 (Multimedia view)-18 (Multimedia view), all share the same general shape. They all represent very slight contributions to the polymer molecular motion that is dominated by the zeroth (which the peanut shapes) and the second (which imparts the tilt). These contributions, be they in the linear or the nonlinear flow regimes, take the form of three-member shish-kebabs. Equations (11)- (20) are the main results of this work. From Figure 19 or Figure 20, we learn that for nonlinear behaviors (be they in nearly time steady shearing flow or the fully nonlinear viscoelastic regime) the orientation distribution is peanut-shaped (called lemniscoidal 40 ). Comparing Figures 4-18 (Multimedia view) with Figures 19-Figure 21, we learn that the lemniscoidal shape of the polymer orientation is imparted by the even harmonics, and specifically by the zeroth and second harmonics, but not by the fourth.
Examining Figure 21, we see that the overall orientation distribution hardly departs from spherical, which is the shape for a liquid at rest. In other words, only very slight departures from this sphericity explain the elastic behavior observed for polymeric liquids for smallamplitude oscillatory shear flow.

IV. CONCLUSION
In this paper, we have deepened our understanding of the molecular motion underlying largeamplitude oscillatory shear flow of polymeric liquids by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We did so for the simplest molecular model, the suspension of non-interacting rigid dumbbells in a Newtonian suspending fluid. We have used three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow, (ii) nonlinear viscoelasticity, and (iii) linear viscoelasticity.
We have learned that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. Further, we find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.